Connections of higher order and product preserving functors

summary:In this paper we consider a product preserving functor $\mathcal F$ of order $r$ and a connection $\Gamma $ of order $r$ on a manifold $M$. We introduce horizontal lifts of tensor fields and linear connections from $M$ to $\mathcal F(M)$ with respect to $\Gamma $. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order